SMALL FLUCTUATIONS IN SYSTEMS WITH MULTIPLE STEADY STATES

The effect of small random perturbations on deterministic systems with multiple steady states is considered. The deterministic systems of interest may have two stable nodes and a saddle point. As parameters vary, two or three of the steady states coalesce. This work is concerned with the long time behavior of the system, when it starts near the deterministic separatrix. The separatrix is surrounded by a tube that may contain two stable steady states. The quantity of basic interest is the conditional probability of first exit from the tube through a specified boundary, conditioned on initial position. Formal asymptotic solutions of the backward equation are constructed. The solutions are obtained by a generalized ″ray method″ and are given in terms of various incomplete special functions. Asymptotic solutions of the equation satisfied by the mean value of the first exit time are also constructed.